Vector Space Methods

Code:

PDEEC0005

Acronym:

VSM

Keywords
Classification	Keyword
OFICIAL	Electrical and Computer Engineering

Instance: 2023/2024 - 1S

Active?	Yes
Responsible unit:	Department of Electrical and Computer Engineering
Course/CS Responsible:	Doctoral Program in Electrical and Computer Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
PDEEC	2	Syllabus	1	-	6	42	162

Teaching language

English

Objectives

This is an advanced course in functional analysis and infinite dimensional optimization, with applications in least-squares estimation, nonlinear programming in Banach spaces, optimization. The repertoire of analytical tools related to linear spaces provides the students with the facility to investigate new theoretical concepts in electrical engineering specialties

Learning outcomes and competences

Understanding and use of mathematical tools into the resolution of engineering problems.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Calculus, Numerical analysis and Linear Algebra

Program

1- An introduction to functional analytic approach to optimization; Finite- versus infinitedimensional
spaces .
2- Normed linear spaces, open and closed sets; convergence; continuity; Banach
spaces, Complete subsets, Quotient spaces, Denseness and Separability
3- Fixed points of transformations on Banach Spaces -- Applications to solutions of
ordinary differential and integral equations
4- Hilbert Spaces -- The Projection Theorem; Orthogonal Complements; Gram-Schmidt
Procedure; Minimum distance to a convex set
5- Hilbert Spaces of random variables and stochastic processes; Least-squares
estimation
6- Dual Spaces. The Hahn-Banach Theorem, with applications to minimum norm
problems
7- Linear operators and adjoints
8- Optimization of functionals -- General results on existence and uniqueness of an optimum
9- Optimization of functionals. Gateaux and Frechet derivatives. Extrema; Euler-
Lagrange equations; Min-Max Theorem in Game Theory.
10- Constrained optimization of functionals: Global theory; Convex-concave functionals,
conjugate functionals, dual optimization problems, Lagrange multipliers, sufficiency;
sensitivity, duality; applications .
11- Constrained Optimization; Equality and Inequality Constraints; Kuhn-Tucker´Theorem in infinite dimensions
12- Other related topics (as time permits)

Mandatory literature

Luenberger, David G.; Optimization by Vector Space methods
Polak, E; Optimization: Algorithms and Consistent Approximations, Springer, New York, 1997. ISBN: 0-387-94971-2
Boyd, S. and Vandenberghe, L; Convex Optimization, Cambridge University Press, 2005. ISBN: 0 521 83378 7
Varaiya, Pravin; Lecture notes on optimization, e-book, http://paleale.eecs.berkeley.edu/~varaiya/papers_ps.dir/NOO.pdf

Teaching methods and learning activities

There will be expository lectures in the end of which a list of problems are proposed. Such lectures are followed by discussion classes to treat problems assigned on the subject.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	20,00
Trabalho escrito	80,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	100,00
Trabalho escrito	50,00
Total:	150,00

Eligibility for exams

90% of the homework with grade  greater or equal to 10.

Calculation formula of final grade

0,80 * homework + 0,20 * class participation

Examinations or Special Assignments

Students will have to do different homeworks that should be returned within a
week after being assigned.

Classification improvement

With an extra project on Optimization.

Observations

Classes can be in Portuguese if no foreigners are enrolled.